Remark 1 Similar problems for Fejér means with respect to Walsh and Vilenkin systems can be found in [1]}, [2]}, [3]} (see also [4]} and [5]}). Our method and estimations of Reisz and Fejér kernels (see Lemmas 1 and 2) do not give possibility to prove even similar strong convergence result as for the case of Fejer means. In particular, for any \(f\in H_{1/2},\)  is it possible to prove the following inequality:
\(\frac{1}{\log n}\overset{n}{\underset{k=1}{\sum }} \frac{\left\Vert R^\alpha _k f\right\Vert _{1/2}^{1/2}}{k}\le c\left\Vert f\right\Vert _{H_{1/2}}^{1/2}, \ \text{where} \ \alpha =w\text{ or }\alpha =\psi ?\) 

